”Extremal Functions on Cayley Digraphs of Finite Cyclic Groups”, in Proceedings of the 2012...

Extremal Functions on Cayley Digraphs of

Finite Cyclic Groups∗

Jordan Blocher

Department of Mathematics

University of Nevada-Reno

Reno, Nevada, USA

[email protected]

Samantha Hampton


University of Arkansas

Fayetteville, Arkansas, USA

[email protected]

Christopher Linden


University of California at Los Angeles

Los Angeles, California, USA

[email protected]

February 5, 2014

∗Supported in part by National Science Foundation (NSF).

Draft printed on February 5, 2014

Blocher, Hampton, Linden, Cayley Digraphs of Finite Cyclic Groups 2

Abstract

For positive integers d and k we define m(d, k) to be the maximum positive integerm such that the Cayley digraph Cay(Zm,A ) of Zm generated by a set A of k positiveintegers has diameter less than or equal to d, where Zm is the finite cyclic group ofresidue classes modulo m. In this paper we present new algorithms that lead to thecomputation of lower bounds for m(d, k) with k small and fixed. These algorithms aredetailed as a complement to the actual proof. We also consider m(2, k) and establish anew lower bound, namely

m(2, k) ≥ 37

121k2 +O(k) as k →∞.

In this paper, we also use this lower bound to obtain a lower bound for m(d, k) for anygiven positive integer d:

m(d, k) ≥(

148

121

)b d2 c(kd

)d

+O(kd−1) as k →∞.



1 Introduction

Let Γ be a finite group with a subset A . The Cayley digraph, denoted Cay(Γ,A ), is a

digraph with vertex set Γ, such that (x, y) is a directed edge if and only if yx−1 ∈ A . The

focus of this paper is the finite group Zm; we will denote the Cayley graph Cay(Zm,A ) as

Cay(m,A ).

0

1

2

3

4

5

6

7

8

9

10

11

Figure 1: Cay(Z12, 1, 6, 11).

Note that Cay(Z12, 1, 6, 11) has the same diameter and degree as the 3-cube, but this

graph contains 12 vertices. It is natural to ask for the maximum number of vertices a Cayley

digraph could possibly have with given diameter and maximum degree. Cayley digraphs

Cay(Zm,A ) of Zm are called circulant (di)graphs. In this paper, we focus on circulant

graphs. However, n-cubes may not be as efficient as circulant graphs of equal degree and

diameter. Cayley digraphs, and circulant graphs in particular are often used as models

for interconnection networks. As such, it is naturally desirable to maximize the number of

vertices which can be connected with constraints on diameter and degree. For any positive

integer d we define

m(d,A ) = maxm|diam(Cay(m,A )) ≤ d,

the largest positive integer m such that the diameter, d(m,A ), of the Cayley digraph

Cay(m,A ) is less than or equal to d. For positive integers d and k,

m(d, k) = maxm(d,A )||A | = k,



the maximum modulus m such that there exists a generating set with cardinality equal to

k and the diameter of the Cayley digraph is less than or equal to d.

It is clear that m(1, k) = k+ 1 and m(d, 1) = d+ 1. In 1974 Wong and Coppersmith [?]

proved that, for positive integers d and k,⌊d

k+ 1

⌋k≤ m(d, k) ≤

(k + d

k

).

To see the the upper bound, let |A | = k such that diam(Cay(Zm,A )) ≤ d. Noting that

every element in Zm is the sum of at most d not necessarily distinct elements of A , and

the commutativity of integers, we see that

m ≤(k + 1 + d− 1

k

)=

(k + d

k

),

thus proving the upper bound. Let

t =

⌊d

k

⌋+ 1

and

A = 1, t, t2, ..., tk−1.

For any x ∈ [0, tk − 1] we have

x =

k−1∑i=1

citi.

Sincek∑i=1

ci ≤ k(t− 1) = k

⌊d

k

⌋≤ d,

we see that d(0, x) ≤ d. Hence, diam(Cay(Ztk ,A ) ≤ d. Thus we see

m(d, k) ≥ tk =

⌊d

k+ 1

⌋k.

Hsu and Jia [?] proved that

m(d, 2) =

⌊d(d+ 4)

3

⌋+ 1 (1)

for all d ≥ 2.



We will begin by examining the case when k = 3. The current bounds for this case, as

d→∞, are176

2197d3 +O(d2) ≤ m(d, 3) ≤ 1

14− 3√

3(d+ 3)3.

2 Algorithm to Construct a Lower Bound m(d, k)

We now provide an algorithm that takes as an input a set of generators and two sets of

polynomial coefficients. These inputs are used to test the ability of a generated lower

bound to provide a set covering for our Cayley Graph. The algorithm iterates through

permutations of the generating set and possible coverings, giving as an output the maximal

generating set as well as the optimal polynomial bound.

The algorithm takes advantage of the fact that there exists a one-to-one correspondance

between representations of the residual classes of Zm and the set of integers [0,m− 1].

2.1 Polynomial Construction

Let d be the diameter for Cay(m,A ).

Our lower bound on m(d, k) will be defined as a polynomial in terms of λ. Note that in the

code, the representative data structures will be ordered from the higher-order to lower-order

polynomial terms. λ is a large number determined by d. The parameter λ will not appear

in the code, but it enables us to compute a lower bound as a function of d.

2.1.1 Permutations of Generators and Coefficients

The permutations tested may be stored as data structures or file tables.

Given d define d1 fixed to be dλ . This will be the d referred to in the algorithm.



The purpose of the following data structures is to provide containers for the following:

Define A to be a set of generators, A = (ai) such that ai = αi+1ai+1λ,∀i ∈ [0, k−1], where

ai a sequence of nonnegative coefficients, and |A | = k.

Define a set of non-negative coefficients c1, c2, .., ck such that ci+1 < αiλ.

Lastly, define a second set of non-negative coefficients x1, x2, .., xk such that xi+1 < d −∑i0 xi. Assume that the set of generator combinations is defined as an upper-triangular

two-dimensional array of unordered coefficients represented by tuples, the sequences of

coefficients are defined as ordered lists of tuples.

Our representative polynomial can now be defined.

m(d, k) = a1c1 + a2c2 · · ·+ akck = (Πk1αi)c1λ

k + (Πk2αi)c2λ

k−1 · · · (αk)ckλ

Tuple Data Structure

template<TP, N>class Tuple

vector<TP> data ; // c l a s s con ta ins an N−vec t o r o f type TP

. . . ;

Construction of the Coefficient Permutations

typedef Tuple<int , k> T;

// Construct ion o f X−Coe f f i c i e n t sfor ( x1 = d ; x1 >= 0 ; −−x1 )

for ( x2 = d − x1 ; x2 >= 0 ; −−x2 )

for ( x3 = d − x1 − x2 ; x3 >= 0 ; −−x3 )

. . .for ( xk = d − x1 − x2 − x3 − . . − xk ; xk >= 0 ; −−xk )

i f ( x1 + x2 + x3 + . . +xk <= d − k )

out << T(xk , xk−1, . . , x1 ) ; // In t h i s case we ares t o r i n g in a data f i l e .



++s i z e ;

. . .

// Construct ion o f M−Coe f f i c i e n t sfor ( int c1 = 1 ; c1 < (dˆk / (k ! ∗ a1 ∗ a2 ∗ . . ∗ ak−1) ) ; ++c1 )

for ( int c2 = 1 ; c2 < a1 ; ++c2 )

for ( int c3 = 1 ; c3 < a2 ; ++c3 )

. . .for ( ck = 1 ; ck < ak−1; ++ck )

out << T( ck , ck−1, . . , c1 ) ;++s i z e ;

. . .

// Construct ion o f Generatorsfor ( a1 = 2 ; a1 < (dˆk / k ! ) ; a1++) // We are us ing the t r i v i a l upper bound to

cons t ra in the s i z e o f the genera to r s .

for ( a2 = 2 ; a2 < (dˆk / k ! ) ; a2++)

. . .i f ( a1 ∗ a2 ∗ . . ∗ ak−1 < dˆk / k ! )

out<< T( a1 ∗ a2 ∗ . . ∗ ak−1, a2 ∗ a3 ∗ . . ak−1, . . . , ak−1, 1);

++s i z e ;

. . .

2.1.2 Polynomial Construction

Our lower bound on m(d, k) will be defined as m(d, k) = aiciλ. To determine the validity

of the lower bound, we compute every point in dA as a polynomial in terms of λ.



Polynomial P( Tuple A, Tuple Y) ;

Polynomial Data Structure

The basic polynomial data structure may be defined as follows:

class Polynomial

Tuple A; // conta iner f o r GeneratorsTuple Y; // conta iner f o r M−Coe f f i c i e n t s or X−Coe f f i c i e n t sint s ub t r a c t i on s [ k ] ;. . .

;

In the case where Polynomial P.Y is a container for the M-Coefficients, the class instance

is a representation of the polynomial x. In the case where P.Y is a container for the X-

Coefficients, the class instance is a representation of X ′. P.Y[0] will refer to the largest

polynomial coefficient. P.A[0] will refer to the largest corresponding generator.

2.1.3 Constructing a Representative Class of Polynomials

For all constructed polynomials x = x1a1 + x2a2 + .. + xkak, we define a representative

x′ ∈ [1,m− 1] to which we will map all congruent polynomials, forming our residue class x

of regular polynomials.

Proposition 2.1. (x1, x2, ..., xk)|x1 ≤ c1, x2 ≤ c2, .., xk ≤ ck, and∑

i xi ≤ d1 define

polynomials x that are considered to be minimal.

Proposition 2.2. The constructed polynomial x is defined to be regular if x ∈ [0,m− 1].

We will use regular polynomials to determine whether or not dA = Zm by only consid-

ering a single covering of Zm.



Note that a regular polynomial need not be minimal.

We check for regularity by comparing the coefficients (x1, x2, .., xk) and (c1, c2, .., ck) from

their respective polynomials.

Proposition 2.3. ∀x ∈ dA if x /∈ [0,m−1], we can identify x with point x′ = (x′1, x′2, .., x

′k) ∈

[0,m − 1] congruent to x (mod m). Then if every point n ∈ Zm is either equal to some x

or x′, then dA ∼= Zm.

Consider the case, x > m(d, k), where x is not regular. We perform a recursive polynomial

subtraction of the coefficients where c1, c2, c3 is subtracted term-by-term from x1, x2, x3, ..., xk.

The resulting low-order coefficients are then forced to be non-negative by adding the gen-

erator associated with the next higher-order term.

The number of times that the representative set cycles Zm as well as the number of times

that a generator needs to be ”borrowed” into the next lowest-order term is saved to use to

create the intervals bounding the representative coefficients in the mathematical proof.

// Overloaded po lynomia l s u b t r a c t i on opera tor .Polynomial Polynomial : : operator−(Polynomial m)

loop :while ( Y > rhs .Y )

Y = T(Y[ 0 ] − rhs .Y[ 0 ] , Y[ 1 ] − rhs .Y[ 1 ] , Y[ 2 ] − rhs .Y[ 2 ] , . . . , Y[ k ]− rhs .Y[ k ] ) ;

++sub t r a c t i on s [ 0 ] ;while ( Y[ 1 ] < 0 )

Y = T(Y[ 0 ] , Y[ 1 ] + (A[ 0 ] / A[ 1 ] ) , Y[ 2 ] , . . . , Y[ k ] ) ;++sub t r a c t i on s [ 1 ] ;

while ( Y[ 2 ] < 0 )

Y = T(Y[ 0 ] , Y[ 1 ] , Y[ 2 ] + (A[ 1 ] / A[ 2 ] ) , . . . , Y[ k ] ) ;++sub t r a c t i on s [ 2 ] ;

while ( Y[ 3 ] < 0 )

Y = T(Y[ 0 ] , Y[ 1 ] , Y[ 2 ] , Y[ 3 ] + (A[ 2 ] / A[ 3 ] ) , . . . , Y[ k ] ) ;++sub t r a c t i on s [ 3 ] ;



. . .

while ( Y[ k−1] < 0 )

Y = T(Y[ 0 ] , Y[ 1 ] , Y[ 2 ] , Y[ 3 ] + (A[ 2 ] / A[ 3 ] ) , . . . , Y[ k−1] + (A[ k−2]/ A[ k−1]) ) ;

++sub t r a c t i on s [ k−1] ;i f ( Y > rhs .Y ) goto loop ;

return ∗ this ;

2.1.4 Construction of the Polynomial Bound

To construct a lower bound, we systematically check combinations of generators A and

coefficients, and record the largest m (and corresponding generators) such that a covering

by dA is achieved.

We will require that our polynomial representative of m to be regular, in order to ensure

x′ ∼= x.

Proposition 2.4. Define x′ to be regular if ∀xi, xi < ci+1.

\\ Def ine a class f unc t i on to return m, the summation o f our polynomial bound .TP Polynomial : : va lue ( )

return ( a1∗y1 + a2∗y2 + + ak∗yk ) ;

\\ Def ine a class f unc t i on to check that x ’ i s r e gu l a rbool Polynomial : : i sRegu la r ( ) const

re turn (Y[ 1 ] < ( A[ 0 ] / A[ 1 ] ) && Y[ 2 ] < (A[ 1 ] / A[ 2 ] ) && . . && Y[ k ] < (A[ k−1] / A[ k ] ) ) ;

Proposition 2.5. ∀n ∈ Zm, ∃x such that x = x′ = n, then dA ⊇ Zm.

We will use a boolean array in order to check the inclusion of Zm in dA . Note that in order

for the polynomial comparison to function properly, the tuple comparisons should be done

in lexigraphical order as for regular tuples, this corresponds to the actual polynomial order.



Polynomial mbest ; // Container f o r our b e s t lower bound m.

vector<bool> cover ;T A; // Container f o r a genera t ing s e t .T Q; // Container f o r a s e t o f M−Coe f f i c i e n t sT x ; // Container f o r a s e t o f X−Coe f f i c i e n t s .// NOTE: The A, Q, and X t u p l e s are r e t r i e v e d from a f i l e or data s t r u c t u r e .

for ( i = 0 ; i < s izeof ( g ene ra to r s ) // Number o f g enera t ing s e t s . ; ++i )

T A(ak , ak−1, . . , a1 ) ;

for ( j = 0 ; j < s izeof ( mcoe f f s ) // Number o f m−c o e f f i c i e n t s . ; ++j )

T Q( ck , ck−1, . . , c1 ) ;

Polynomial M(A, Q) ; // Create i n i t i a l po lynomia l r e p r e s en t a t i on o fthe lower bound .

cover . c l e a r ( ) ;i f ( (M. value ( ) > mbest . va lue ( ) ) ) // Ignore M tha t are too sma l l .

for ( k = 0 ; k < s izeof ( x c o e f f s ) // Number o f x−c o e f f i c i e n t s . ; ++k )

T x(xk , xk−1, . . , x1 ) ;

Polynomial X(A, x ) ; // Create our i n i t i a l r e p r e s e n t a t i v e .Polynomial X prime (X−M) ; // Enforce r e g u l a r i t y .i f ( X prime . wellFormed ( ) )

cover . push back (1 ) ;

i f ( accumulate ( cover . begin ( ) , cover . end ( ) ,0 ) == M. value ( ) )

mbest = M; // Store improved lower bound .

Parallelization

The looping nature of the program makes it a natural candidate for optimization using

parallel processing techniques. In this case, we have used a cluster of computer nodes. The

largest dataset that is iterated over is the M-Coefficients, and can be a ”bottleneck” in the

program. Parallelization solves this problem by allowing a number of nodes that compute



extremely large datasets to compute for an extended time or even without completion.

These larger computations will not interfere with nodes that are computing bounds based

on subsequent generating sets. The effect of the multiple processes is that we are able to

check a larger set of generators, in fact checking up to the current trivial lower bound of dk

k! .

2.2 Complexity of the Algorithm

2.2.1 Generators and Coefficients

The complexity of the algorithm depends both on the number of generators as well as

the chosen diameter, however since our program deals with smaller values of k, and our

chosen diameter may be arbitrarily large, we will formulate our complexity analysis in

terms of d. As we are checking all permutations of generators up to the upper bound dk

k! ,

we are able to treat the coefficient 1(k!)k

as a constant term and express the compexity as

an average case in big-O notation. The number of operations per loop in the construction

of the generators is on the order of O(dk). The construction of the coefficients to generate

our representative polynomial m is dependent on the individual generating set for which

it is being constructed. In the worst case, the generating set will have large generators,

allowing for the outermost loop to finish quickly, however, this will make it possible for

the inner loops to approach or surpass the complexity of the larger, outermost loop. The

complexity in this case is O(maxa∈A (a ·d)k), where for most a ∈ A , a < d. In the best case,

the generating set will have small coefficients, in which case the outermost loop will have

complexity of Ω(maxa∈A (da)k), and the inner loops will finish quickly. The final complexity

of the loop structure in this case can be described as Θ(dk). Finally, the coefficients for our

representative polynomial are generated in O(dk).

2.2.2 Polynomial Bound

The generation of the polynomial bound is again a nested loop structure which checks each

possible representative for each possible combination for each generating set that can be

created where each generator is the largest possible multiple it can take for its position

up to the trivial upper bound. We can describe the complexity of this main program as

O(dk)·Θ(dk)·O(dk) = O(dk3). As we found, the size of the coefficient set for the polynomial



bound depends largely on the specific generating set being examined, significantly affecting

the complexity of the inner loops (or coefficient loops).



3 A New Lower Bound for m(2, k)

Let d and k be positive integers. Let n(d, k) denote the largest positive integer n such that

there exists a subset A of k positive integers with the property that every integer in the

interval [0, n] is the sum of at most d not necessarily distinct elements of A . In other words,

n(d, k) = maxA⊆Z+

|A |=k

n | dA ⊇ [0, n].

Recall that dA denotes the set of all sums of at most d not necessarily distinct elements of

A . The computation of n(d, k) is often referred as the postage stamp problem. The postage

stamp problem is an old problem in number theory that has been studied extensively. See

[?, ?, ?] for more information. A simple construction shows that

n(2, k) ≥ 1

4k2 +O(k).

Rohrbach [?] conjectured in 1937 that

n(2, k) ∼ 1

4k2, (2)

Hammerer and Hofmeister [?] proved in 1976 by an explicit construction of a 2-basis that

n(2, k) ≥ 5

18k2 +O(k),

which disproves the conjecture of Rohrbach (??). The best known lower bound for n(2, k)

was proved by Mrose [?]

n(2, k) ≥ 2

7k2 +

12

7k +O(1) as k →∞. (3)

If dA ⊇ [0, n] then dA = Zn+1. Therefore, for all positive integers d and k we have

m(d, k) ≥ n(d, k) + 1.

This implies the best known lower bound for m(2, k) in (??) before our result in this paper.

Our proof uses a similar construction to one used by Torleiv Klove and Mossige used to

prove the above lower bound on n(2, k) in (??). We used a computer program to test

candidates for the values that appeared in the proof as the subscripts µ, ν and η. It is

likely that our result could be improved upon by a more complicated construction using a

larger number of the same types of subsets. However, testing larger constructions becomes

computationally difficult, and there is a theoretical limit of 1/3 on the leading coefficient of

such a construction. Any lower bound better than1

3k2 +O(k) would be of interest.



Theorem 3.1. m(2, k) ≥ 37

121k2 +O(k) as k →∞.

et k ≥ 14 be an integer. Let k1 =

⌊k − 3

11

⌋. Let m = 37k21. Define

Iµ = [µk21, µk21 + k1], µ = 0, 4, 15, 26;

Sν = νk21 + ik1 | i = 0, 1, ..., k1 − 1, ν = 0, 1, 2, 3;

Tη = ηk21 + i(k1 + 1) | i = 0, 1, ..., k1 − 1, η = 10, 20, 30.

Let S = S0 ∪ S1 ∪ S2 ∪ S3, and define

A = I0 ∪ I4 ∪ I15 ∪ I26 ∪ S ∪ T10 ∪ T20 ∪ T30.

Noting that I0 ∩ S0 = 0, and

|Iµ| = k1 + 1, |Sν | = k1, and |Tη| = k1,

we see that

|A | ≤ 11k1 + 3 ≤ k.

We now prove that A +A = Zm. We begin by claiming Iµ+Tη ⊇ [(µ+η)k21, (µ+η+1)k21).

Let n ∈ [(µ+ η)k21, (µ+ η + 1)k21). Then we can write n as

n = (µ+ η)k21 + qk1 + r,

where 0 ≤ q < k1 and 0 ≤ r < k1.

If r ≥ q, then 0 ≤ r − q < k1 and

n = µk21 + (r − q) + ηk21 + q(k1 + 1).

Since

µk21 + (r − q) ∈ Iµ and ηk21 + q(k1 + 1) ∈ Tη,

we see that n ∈ Iµ + Tη.

If r < q, then we must have q ≥ 1 and 0 ≤ k1 + r − q + 1 ≤ k1. Then

µk21 + (k1 + r − q + 1) ∈ Iµ and ηk21 + (q − 1)(k1 + 1) ∈ Tη.

Hence

n = µk21 + (k1 + r − q + 1) + ηk21 + (q − 1)(k1 + 1) ∈ Iµ + Tη.



Next we claim that Iµ+Sν ⊇ [(µ+ν)k21, (µ+ν+1)k21). Let n ∈ [(µ+ν)k21, (µ+ν+1)k21).

Then we can write n as

n = (µ+ ν)k21 + qk1 + r = µk21 + r + νk21 + qk1,

where 0 ≤ q < k1 and 0 ≤ r < k1, such that

µk21 + r ∈ Iµ and νk21 + qk1 ∈ Sν ,

so n ∈ Iµ + Sν .

Our final claim is S + Tη ⊇ [(η + 1)k21, (η + 4)k21). Let n ∈ [(η + 1)k21, (η + 4)k21). Then

we can write n as

n = (η + ν)k21 + qk1 + r,

where 1 ≤ ν ≤ 3, 0 ≤ q < k1, and 0 ≤ r < k1.

If q ≥ r, then

n = νk21 + (q − r)k1 + ηk21 + r(k1 + 1),

where νk21 + (q − r)k1 ∈ Sν and ηk21 + d(k1 + 1) ∈ Tη, so n ∈ Sν + Tη.

If q < r, then

n = (ν − 1)k21 + (k1 + q − r)k1 + ηk21 + r(k1 + 1),

where

(ν − 1)k21 + (k1 + q − r)k1 ∈ Sν−1 and ηk21 + r(k1 + 1) ∈ Tη.

Hence n ∈ Sν−1 + Tη ⊆ S + Tη. It is clear that, in Z37k21,

[45k21, 46k21) = [8k21, 9k21),

[46k21, 47k21) = [9k21, 10k21),

[56k21, 57k21) = [19k21, 20k21).



Therefore, we have proved that the entire interval [0,m) = Zm can be covered as follows:

I0 + S ⊇ [0, 4k21),

I4 + S ⊇ [4k21, 8k21),

I15 + T30 ⊇ [45k21, 46k21) = [8k21, 9k21),

I26 + T20 ⊇ [46k21, 47k21) = [9k21, 10k21),

I0 + T10 ⊇ [10k21, 11k21),

S + T10 ⊇ [11k21, 14k21),

I4 + T10 ⊇ [14k21, 15k21),

I15 + S ⊇ [15k21, 19k21),

I26 + T30 ⊇ [56k21, 57k21) = [19k21, 20k21),

I0 + T20 ⊇ [20k21, 21k21),

S + T20 ⊇ [21k21, 24k21),

I4 + T20 ⊇ [24k21, 25k21),

I15 + T10 ⊇ [25k21, 26k21),

I26 + S ⊇ [26k21, 30k21),

I0 + T30 ⊇ [30k21, 31k21),

S + T30 ⊇ [31k21, 34k21),

I4 + T30 ⊇ [34k21, 35k21),

I15 + T20 ⊇ [35k21, 36k21),

I26 + T10 ⊇ [36k21, 37k21).

It now follows that

A + A ⊇ [0, 37k21).



Hence

m(2, k) ≥ 37k21

= 37 ·⌊k − 2

11

⌋2> 37

(k − 2

11− 1

)2

=37

121k2 − 962

121k +

6253

121

=37

121k2 +O(k) as k →∞.

Theorem ?? is proved.

4 Extension to the General Case

Theorem 4.1. For any integer d ≥ 2, as k →∞, we have

m(d, k) ≥(

148

121

)b d2c(k

d

)d+O(kd−1).

Before we start the proof of Theorem ??, we state and prove the following addition

lemma for m(d, k).

Lemma 4.1. Given positive integers k1, k2, d1 ≥ 2 and d2 ≥ 2, we have

m(d1 + d2, k1 + k2) ≥ m(d1, k1)m(d2, k2)

Proof of Lemma ??. Let A1 and A2 be sets of positive integers such that m(d1,A1) =

m(d1, k1) and m(d2,A2) = m(d2, k2). Since d1, d2 ≥ 2 we have that m(d1, k1) is greater

than every element in A1, and similarly for m(d2, k2) and A2. So let A = A1 ∪m(d1, k1) ·x | x ∈ A2. Then it suffices to show that A is a (d1 + d2)-basis for Zm(d1,k1)m(d2,k2).

We do this by writing n ∈ Zm(d1,k1)m(d2,k2) as xm(d1, k1) + r where x ∈ Zm(d2,k2) and

r ∈ Zm(d1,k1). Then xm(d1, k1) ∈ d1A1m(d1, k1) · x | x ∈ A2 and r ∈ d2A1, so

n ∈ (d1 + d2)A .



Proof of Theorem ??. Let d = 2q + r where q =

⌊d

2

⌋and k = du+ v where u =

⌊k

d

⌋. We

separate the calculation into two cases. Case 1: If r = 0, then

m(d, k) = m(2q, du+ v)

= m(2q, (2q)u+ v)

≥ m(2q, 2qu) ≥ m(2, 2u)q

≥(

37

121(2u)2 +O(u)

)q=

(148

121

(k − vd

)2

+O(u)

) d2

=

(148

121

) d2(k

d

)d+O(kd−1)

Case 2: If r = 1, then

m(d, k) = m(2q + 1, du+ v)

= m(2q + 1, (2q + 1)u+ v)

≥ m(2q + 1, 2qu+ u) ≥ m(2, 2u)q ·m(1, u)

≥(

37

121(2u)2 +O(u)

)q· (u+ 1)

=

(148

121u2 +O(u)

) d−12

· (u+ 1)

=

(148

121

d−12

ud−1 +O(ud−2)

)· (u+ 1)

=148

121

d−12

ud +O(ud−1)

=

(148

121

) d−12(k

d

)d+O(kd−1)

Hence

m(d, k) ≥(

148

121

)b d2c(k

d

)d+O(kd−1).

The proof of Theorem ?? is complete.



5 A Lower Bound of m(d, 4).

In 1974, Wong and Coppersmith proved that

m(d, k) ≥(d

k+ 1

)k.

In the case when k = 4, this gives the lower bound to be

m(d, k) ≥(d

4

)4

+O(d3).

Jia later improved this lower bound, and proved that

m(d, 4) ≥ 2.048

(d

4

)4

+O(d3) as d→∞.

In 1992, Chen and Gu proved that

m(d, 4) ≥ 3.2768

(d

4

)4

+O(d3) as d→∞.

In this paper we will show that

m(d, 4) ≥ 512

243

(d

4

)4

+O(d3) ≈ 2.106996

(d

4

)4

+O(d3).

This is not an improvement on the 1992 bound, but is instead given as an example of a

proof of a lower bound derived from our computational algorithm.

Theorem 5.1. As d→∞,

m(d, 4) ≥ 512

243

(d

4

)4

+O(d3) ≈ 2.106996

(d

4

)4

+O(d3).

Proof. Let d ≥ 11 be an integer and let λ =⌊d−29

⌋. Define

α = 3λ,

β = 3λα,

γ = 3λβ,

m = 2λγ + λβ + λα+ λ.



Let A = 1, α, β, γ. Then

m = 2λd+ λc+ λb+ λ

= 54λ4 + 9λ3 + 3λ2 + λ

=512

243

(d

4

)4

+O(d3).

Let dA denote the set of all sums of at most d not necessarily distinct elements of a

generating set A of Zm. Then for this proof we need to show that dA = Zm such that the

Cayley digraph Cay(m,A) has diameter d(m,A) ≤ d.

Every integer n such that 0 ≤ n < m can be expressed in the following way:

n = w + xα+ yβ + zγ

where

0 ≤ w ≤ 3λ, 0 ≤ x ≤ 3λ, 0 ≤ y ≤ 3λ, 0 ≤ z ≤ 2λ.

Thus we only need to show, for every 0 ≤ n < m, there exists nonnegative integers δ1, δ2,

δ3, and δ4 such that we can write n as

n ≡ δ1 + δ2α+ δ3β + δ4γ

where

δ1 + δ2 + δ3 + δ4 ≤ d.

We now consider the following cases:

Case 1. 0 ≤ x3 < λ. The following subcases need to be considered:

Subcase 1.a. If 2λ ≤ x1 < 3λ then 0 ≤ x0 < 2λ.

If 0 ≤ x1 < 2λ, we have

x0 + x1 + x2 + x3 ≤ 3λ+ 2λ+ 3λ+ λ = 9λ ≤ d,

which implies that n = x0 + x1α+ x2β + x3γ ∈ dA.

If 2λ ≤ x1 < 3λ and 0 ≤ x0 < 2λ, we have

x0 + x1 + x2 + x3 ≤ 2λ+ 3λ+ 3λ+ λ = 9λ ≤ d,




Subcase 1.b. 2λ ≤ x2 < 3λ, 2λ ≤ x1 < 3λ, 2λ ≤ x0 < 3λ.

We have

n ≡ n+m = x0 + x1α+ x2β + x3γ + λ+ λα+ λβ + 2λγ

= x0 + λ+ (x1 + λ)α+ (x2 + λ)β + (x3 + 2λ)γ

= x0 − 2λ+ (x1 − 2λ+ 1)α+ (x2 + λ+ 1)β + (x3 + 2λ)γ.

Noting that

x0 − 2λ ≥ 0, x1 − 2λ+ 1 ≥ 0, x2 + λ+ 1 ≥ 0, and x3 + 2λ ≥ 0,

we see that

x0 − 2λ+ x1 − 2λ+ 1 + x2 + λ+ 1 + x3 + 2λ = x0 + x1 + x2 + x3 − λ+ 2

≤ 3λ+ 3λ+ 3λ+ λ− λ+ 2

= 9λ+ 2 ≤ d,


Case 2. λ ≤ x3 < 2λ. The following subcases need to be considered:

Subcase 2.a. 0 ≤ x2 < 2λ. If 2λ ≤ x1 < 3λ then 0 ≤ x0 < 2λ.


x0 + x1 + x2 + x3 ≤ 2λ+ 2λ+ 2λ+ 3λ = 9λ ≤ d,



x0 + x1 + x2 + x3 ≤ 2λ+ 2λ+ 3λ+ 2λ = 9λ ≤ d,


Subcase 2.b. λ ≤ x2 < 2λ, 2λ ≤ x1 < 3λ, 2λ ≤ x0 < 3λ.

We have


= x0 + λ+ (x1 + λ)α+ (x2 + λ)β + (x3 + 2λ)γ

= x0 − 2λ+ (x1 − 2λ+ 1)α+ (x2 + λ+ 1)β + (x3 + 2λ)γ.



Noting that

x0 − 2λ ≥ 0, x1 − 2λ+ 1 ≥ 0, x2 + λ+ 1 ≥ 0, and x3 + 2λ ≥ 0,

we see that

x0 − 2λ+ x1 − 2λ+ 1 + x2 + λ+ 1 + x3 + 2λ = x0 + x1 + x2 + x3 − λ+ 2

≤ 3λ+ 3λ+ 2λ+ 2λ− λ+ 2

= 9λ+ 2 ≤ d,


Subcase 2.c. 2λ ≤ x2 < 3λ. If λ ≤ x1 < 2λ then 0 ≤ x0 < 2λ.

If 0 ≤ x1 < λ, we have

x0 + x1 + x2 + x3 ≤ 3λ+ λ+ 3λ+ 2λ = 9λ ≤ d,


If λ ≤ x1 < 2λ and 0 ≤ x0 < 2λ, we have

x0 + x1 + x2 + x3 ≤ 2λ+ 2λ+ 3λ+ 2λ = 9λ ≤ d,


Subcase 2.d. 2λ ≤ x2 < 3λ, λ ≤ x1 < 2λ, 2λ ≤ x0 < 3λ.

We have


= x0 + λ+ (x1 + λ)α+ (x2 + λ)β + (x3 + 2λ)γ

= x0 − 2λ+ (x1 + λ+ 1)α+ (x2 − 2λ)β + (x3 + 2λ+ 1)γ.

Noting that

x0 − 2λ ≥ 0, x1 + λ+ 1 ≥ 0, x2 − 2λ ≥ 0, and x3 + 2λ+ 1 ≥ 0,

we see that

x0 − 2λ+ x1 + λ+ 1 + x2 − 2λ+ x3 + 2λ+ 1 = x0 + x1 + x2 + x3 − λ+ 2

≤ 3λ+ 2λ+ 3λ+ 2λ− λ+ 2

= 9λ+ 2 ≤ d,




Subcase 2.e. 2λ ≤ x2 < 3λ, 2λ ≤ x1 < 3λ, 0 ≤ x0 < 2λ.

We have


= x0 + λ+ (x1 + λ)α+ (x2 + λ)β + (x3 + 2λ)γ

= x0 + λ+ (x1 − 2λ)α+ (x2 − 2λ+ 1)β + (x3 + 2λ+ 1)γ.

Noting that

x0 + λ ≥ 0, x1 − 2λ ≥ 0, x2 − 2λ+ 1 ≥ 0, and x3 + 2λ+ 1 ≥ 0,

we see that

x0 + λ+ x1 − 2λ+ x2 − 2λ+ 1 + x3 + 2λ+ 1 = x0 + x1 + x2 + x3 − λ+ 2

≤ 2λ+ 3λ+ 3λ+ 2λ− λ+ 2

= 9λ+ 2 ≤ d,


Subcase 2.f. 2λ ≤ x2 < 3λ, 2λ ≤ x1 < 3λ, 2λ ≤ x0 < 3λ.

We have


= x0 + λ+ (x1 + λ)α+ (x2 + λ)β + (x3 + 2λ)γ

= x0 − 2λ+ (x1 − 2λ+ 1)α+ (x2 − 2λ+ 1)β + (x3 + 2λ+ 1)γ.

Noting that

x0 − 2λ ≥ 0, x1 − 2λ+ 1 ≥ 0, x2 − 2λ+ 1 ≥ 0, and x3 + 2λ+ 1 ≥ 0,

we see that

x0 − 2λ+ x1 − 2λ+ 1 + x2 − 2λ+ 1 + x3 + 2λ+ 1 = x0 + x1 + x2 + x3 − 4λ+ 3

≤ 3λ+ 3λ+ 3λ+ 2λ− 4λ+ 3

= 7λ+ 2 ≤ d,




Case 3. 2λ ≤ x3 < 3λ. The following subcases need to be considered:

Subcase 3.a. 0 ≤ x2 < λ. If 2λ ≤ x1 < 3λ then 0 ≤ x0 < 2λ.


x0 + x1 + x2 + x3 ≤ 3λ+ 2λ+ λ+ 3λ = 9λ ≤ d,



x0 + x1 + x2 + x3 ≤ 2λ+ 3λ+ λ+ 3λ = 9λ ≤ d,


Subcase 3.b. 0 ≤ x2 < λ, 2λ ≤ x1 < 3λ, 2λ ≤ x0 < 3λ.

We have


= x0 + λ+ (x1 + λ)α+ (x2 + λ)β + (x3 + 2λ)γ

= x0 − 2λ+ (x1 − 2λ+ 1)α+ (x2 + λ+ 1)β + (x3 + 2λ)γ.

Noting that

x0 − 2λ ≥ 0, x1 − 2λ+ 1 ≥ 0, x2 + λ+ 1 ≥ 0, and x3 + 2λ ≥ 0,

we see that

x0 − 2λ+ x1 − 2λ+ 1 + x2 + λ+ 1 + x3 + 2λ = x0 + x1 + x2 + x3 − λ+ 2

≤ 3λ+ 3λ+ λ+ 3λ− λ+ 2

= 9λ+ 2 ≤ d,


Subcase 3.c. λ ≤ x2 < 2λ and 0 ≤ x1 < 2λ. If λ ≤ x1 < 2λ then 0 ≤ x0 < λ.

If 0 ≤ x1 < λ, we have

x0 + x1 + x2 + x3 ≤ 3λ+ λ+ 2λ+ 3λ = 9λ ≤ d,


If λ ≤ x1 < 2λ and 0 ≤ x0 < λ, we have

x0 + x1 + x2 + x3 ≤ λ+ 2λ+ 2λ+ 3λ = 8λ ≤ d,




Thus we have shown that every element 0 ≤ n < m is contained in dA, which implies

the diameter of the Cayley digraph Cay(Zm, A) is less than or equal to d. Hence,

m(d, 4) ≥ 2.048

(d

4

)4

+O(d3) as d→∞.

6 Open Problems

1. Finding a non-trivial lower bound for m(3, k), m(4, k), etc.

2. For n(d, k) it is known that for every positive integer k the limit

limd→∞

n(d, k)

dk

exists, and the value is known for k = 1, 2 and 3. It is not known whether or not

limd→∞

m(d, k)

dk

exists for every k, and the value is only known for k = 1 and 2.

3. The limits

limd→∞

m(d, k)

n(d, k)and lim

k→∞

m(d, k)

n(d, k)

are also of interest, if they exist.

4. We may define the undirected version of our extremal function M(d, k) to be the largest

M such that there exists of symmetric set A of k elements and their additive inverses

such that diam(Cay(ZM ,A )) ≤ d. Little is known about M(d, k), for fixed d and k ≥ 3.

5. We may also define the average version of our extremal function m(d, k) to be the largest

m such that there exists a set A of k elements such that the average distance between

any two vertices in Cay(Zm,A ) is less than d. New lower bounds for k ≥ 3 would be

interesting for this function as well.

6. Although this paper is primarily concerned with lower bounds, upper bounds on m(d, k)

(and related functions) are also of interest.


”Extremal Functions on Cayley Digraphs of Finite Cyclic Groups”, in Proceedings of the 2012...

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